of astronomers. Human life may be subject at different ages to a succession of different influences incapable of reduction under any one law. The results observed may in fact be aggregates of an immense number of separate results each governed by its own separate laws, so that the subjects may be complicated beyond the possibility of complete resolution by empirical methods. This is certainly true of the mathematical functions which must some time or other be introduced into the science of political economy.
Simple Proportional Variation.
When we first treat numerical results in any novel kind of investigation, our impression will probably be that one quantity varies in simple proportion to another, so as to obey the law y = mx + n. We must learn to distinguish carefully between the cases where this proportionality is really, and where it is only apparently true. In considering the principles of approximation we found that a small portion of any curve will appear to be a straight line. When our modes of measurement are comparatively rude, we must expect to be unable to detect the curvature. Kepler made meritorious attempts to discover the law of refraction, and he approximated to it when he observed that the angles of incidence and refraction if small bear a constant ratio to each other. Angles when small are nearly as their sines, so that he reached an approximate result of the true law. Cardan assumed, probably as a mere guess, that the force required to sustain a body on an inclined plane was simply proportional to the angle of elevation of the plane. This is approximately the case when the angle is small, but in reality the law is much more complicated, the power required being proportional to the sine of the angle. The early thermometer-makers were unaware whether the expansion of mercury was proportional or not to the heat communicated to it, and it is only in the present century that we have learnt it to be not so. We now know that even gases obey the law of uniform expansion by heat only in an approximate manner. Until reason to the contrary is shown, we should do well to look upon every law of simple proportion as only provisionally true.